# Problems in General Physics → Electrodynamics → Electric Capacitance. Energy of an Electric Field

 3.101. Find the capacitance of an isolated ball-shaped conductor of radius R1 surrounded by an adjacent concentric layer of dielectric with permittivity ε and outside radius R2. Free solution >> 3.102. Two parallel-plate air capacitors, each of capacitance C, were connected in series to a battery with emf ξ. Then one of the capacitors was filled up with uniform dielectric with permittivity ε. How many times did the electric field strength in that capacitor decrease? What amount of charge flows through the battery? Free solution >> 3.103. The space between the plates of a parallel-plate capacitor is filled consecutively with two dielectric layers 1 and 2 having the thicknesses d1 and d2 and the permittivities ε1 and ε2 respectively. The area of each plate is equal to S. Find: (a) the capacitance of the capacitor; (b) the density σ' of the bound charges on the boundary plane if the voltage across the capacitor equals V and the electric field is directed from layer 1 to layer 2. Free solution >> 3.104. The gap between the plates of a parallel-plate capacitor is filled with isotropic dielectric whose permittivity ε varies linearly from ε1 to ε2 (ε2 > ε1) in the direction perpendicular to the plates. The area of each plate equals S, the separation between the plates is equal to d. Find: (a) the capacitance of the capacitor; (b) the space density of the bound charges as a function of ε if the charge of the capacitor is q and the field E in it is directed toward the growing ε values. Free solution >> 3.105. Find the capacitance of a spherical capacitor whose electrodes have radii R1 and R2 > R1 and which is filled with isotropic dielectric whose permittivity varies as ε = a/r, where a is a constant, and r is the distance from the centre of the capacitor. Free solution >> 3.108. Two long straight wires with equal cross-sectional radii a are located parallel to each other in air. The distance between their axes equals b. Find the mutual capacitance of the wires per unit length under the condition b >> a. Free solution >> 3.109. Along straight wire is located parallel to an infinite conducting plate. The wire cross-sectional radius is equal to a, the distance between the axis of the wire and the plane equals b. Find the mutual capacitance of this system per unit length of the wire under the condition a << b. Free solution >> 3.110. Find the capacitance of a system of two identical metal balls of radius a if the distance between their centres is equal to b, with b >> a. The system is located in a uniform dielectric with permittivity ε. Free solution >> 3.111. Determine the capacitance of a system consisting of a metal ball of radius a and an infinite conducting plane separated from the centre of the ball by the distance l if l >> a. Free solution >> 3.112. Find the capacitance of a system of identical capacitors between points A and B shown in (a) Fig. 3.17a; (b) Fig. 3.17b. Free solution >> 3.114. A capacitor of capacitance C1 = 1.0 μF withstands the maximum voltage V1 = 6.0 kV while a capacitor of capacitance C2 = 2.0 μF, the maximum voltage V2 = 4.0 kV. What voltage will the system of these two capacitors withstand if they are connected in series? Free solution >> 3.115. Find the potential difference between points A and B of the system shown in Fig. 3.19 if the emf is equal to ξ = 110 V and the capacitance ratio C2/C1 = η = 2.0. Free solution >> 3.116. Find the capacitance of an infinite circuit formed by the repetition of the same link consisting of two identical capacitors, each with capacitance C (Fig. 3.20). Free solution >> 3.117. A circuit has a section AB shown in Fig. 3.21. The emf of the source equals ξ = 10 V, the capacitor capacitances are equal to C1 = 1.0 μF and C2 = 2.0 μF, and the potential difference φA - φB = 5.0 V. Find the voltage across each capacitor. Free solution >> 3.118. In a circuit shown in Fig. 3.22 find the potential difference between the left and right plates of each capacitor. Free solution >> 3.120. Determine the potential difference φA - φB between points A and B of the circuit shown in Fig. 3.23. Under what condition is it equal to zero? Free solution >> 3.121. A capacitor of capacitance C1 = 1.0 μF charged up to a voltage V = 110 V is connected in parallel to the terminals of a circuit consisting of two uncharged capacitors connected in series and possessing the capacitances C2 = 2.0 μF and C3 = 3.0 μF. What charge will flow through the connecting wires? Free solution >> 3.122. What charges will flow after the shorting of the switch Sw in the circuit illustrated in Fig. 3.24 through sections 1 and 2 in the directions indicated by the arrows? Free solution >> 3.123. In the circuit shown in Fig. 3.25 the emf of each battery is equal to ξ = 60 V, and the capacitor capacitances are equal to C1 = 2.0 μF and C2 = 3.0 μF. Find the charges which will flow after the shorting of the switch Sw through sections 1, 2 and 3 in the directions indicated by the arrows. Free solution >> 3.124. Find the potential difference φA - φB between points A and B of the circuit shown in Fig. 3.26. Free solution >> 3.127. Determine the interaction energy of the point charges located at the corners of a square with the side a in the circuits shown in Fig. 3.29. Free solution >> 3.129. A point charge q is located at a distance l from an infinite conducting plane. Find the interaction energy of that charge with chose induced on the plane. Free solution >> 3.131. A capacitor of capacitance C1 = 1.0 μF carrying initially a voltage V = 300 V is connected in parallel with an uncharged capacitor of capacitance C2 = 2.0 μF. Find the increment of the electric energy of this system by the moment equilibrium is reached. Explain the result obtained. Free solution >> 3.132. What amount of heat will be generated in the circuit shown in Fig. 3.30 after the switch Sw is shifted from position 1 to position 2? Free solution >> 3.134. A system consists of two thin concentric metal shells of radii R1 and R2 with corresponding charges q1 and q2. Find the self-energy values W1 and W2 of each shell, the interaction energy of the shells W12, and the total electric energy of the system. Free solution >> 3.135. A charge q is distributed uniformly over the volume of a ball of radius R. Assuming the permittivity to be equal to unity, find: (a) the electrostatic self-energy of the ball; (b) the ratio of the energy W1 stored in the ball to the energy W2 pervading the surrounding space. Free solution >> 3.136. A point charge q = 3.0 μC is located at the centre of a spherical layer of uniform isotropic dielectric with permittivity ε = 3.0. The inside radius of the layer is equal to a = 250 mm, the outside radius is b = 500 mm. Find the electrostatic energy inside the dielectric layer. Free solution >> 3.137. A spherical shell of radius R1 with uniform charge q is expanded to a radius R2. Find the work performed by the electric forces in this process. Free solution >> 3.138. A spherical shell of radius R1 with a uniform charge q has a point charge q0 at its centre. Find the work performed by the electric forces during the shell expansion from radius R1 to radius R2. Free solution >> 3.139. A spherical shell is uniformly charged with the surface density σ. Using the energy conservation law, find the magnitude of the electric force acting on a unit area of the shell. Free solution >> 3.141. Each plate of a parallel-plate air capacitor has an area S. What amount of work has to be performed to slowly increase the distance between the plates from x1 to x2 if (a) the capacitance of the capacitor, which is equal to q, or (b) the voltage across the capacitor, which is equal to V, is kept constant in the process? Free solution >> 3.142. Inside a parallel-plate capacitor there is a plate parallel to the outer plates, whose thickness is equal to η = 0.60 of the gap width. When the plate is absent the capacitor capacitance equals C = 20 nF. First, the capacitor was connected in parallel to a constant voltage source producing V = 200 V, then it was disconnected from it, after which the plate was slowly removed from the gap. Find the work performed during the removal, if the plate is (a) made of metal; (b) made of glass. Free solution >> 3.143. A parallel-plate capacitor was lowered into water in a horizontal position, with water filling up the gap between the plates d = 1.0 mm wide. Then a constant voltage V = 500 V was applied to the capacitor. Find the water pressure increment in the gap. Free solution >>